In this message, I focus on one passage in Harvey Friedman's,
"Re re re re Harvey on 'a very exciting claim.'" (29 Mar)
> Here are just two famous examples. I saw a book listing more examples,
> and I know that there are some FOM subscribers reading this who can
> help us with mentioning more of them.
> 1. Roth's theorem about approximability of irrational algebraic real
> numbers by rationals.
> Roth was awarded the Fields Medal for his proof of this theorem. No
> constructive proof is known. No effective bounds are known.
> I know that most leading number theorists are very much interested
> in rectifying this situation.
I don't believe it.
> ....Faltings was awarded the Fields Medal for his proof of this
> theorem. No constructive proof is known. No effective bounds are
> known.
>I know that most leading number theorists are very much interested
> in rectifying this situation.
I don't believe it.
More generally, I don't believe that any number theorist, leading
or following, is "very much interested in rectifying this situation."
Yes, they sometimes talk this way. But that has more to do with the
grip of their metaphysics than with number theory.
I would change my mind if I was shown, in at least one case, a
sound mathematical reason for wanting such a constructive proof or
bound. A list of number theorists making pronouncements about this
is surely no substitute for sound mathematical reasons.
I don't know most leading number theorists. But I know some.
Several decades ago, together with Birch and Coates, I supervised
a Ph.D. thesis on curves of conductor 11. During that time, I was
surprised to learn of cases in which an eminent number theorist had
marveled about how somebody had proved that some number (e.g., a sign
change) exists but didn't know how to find it--which had the effect
of making it a challenge to get a bound for it.
The reason the number theorist marveled is that "proving it exists
without being able to find it" sometimes makes it seem as if we have
"a pipeline to God." We don't need constructions to get at the truth!
But this is metaphysics not number theory.
In one case, after a bound for a sign change was hacked out, an
eminent number theorist described it as "the biggest number for
which mathematicians have a use." Use? Use?? They made no use of
it. They just marveled at how big (grotesque?) it was.
Probably to be continued.
Gabriel Stolzenberg